Anamorphic images with rotational symmetry

In our garden we have some polished spheres made of stainless steel. I like to look at them and to see the distorted images of the surrounding. The straight lines of the house and tree trunks become graceful curves. Polished cylinders and pyramids with a circle base make too interesting reflected images. But with a computer we can make even more kinds of anamorphic images. Most of them could not be the result of physical optics.

To get anamorphic images I do similarly as for the kaleidoscopes. For every pixel of the anamorphic picture I have a pair of (x,y)-coordinates. They give the place in the original image, where the color for this pixel has to be read out. First, the coordinates are set equal to the place in the anamorphic picture. Then I apply mappings of the plane to the coordinates. This results in the anamorphosis.

We can use this idea to get pictures with n-fold rotational symmetry. Doing similarly as for the kaleidoscope and thus cutting pieces out of the original image and copying would not do well because there would be discontinuous cuts. But a simple anamorphosis does well if we use the mathematics of the post “Fractal surprise from complex function iteration“. We look at the complex number z=x+iy, that corresponds to the coordinates. The n-th power of z results in an n-fold rotational symmetry, as it multiplies the angle between the x-axis and line from the origin and the point (x,y) by n.

Here is an example with three-fold rotational symmetry. I have used a photo of cars in front of our post office. False colors result in:

many carsCorrExtending this idea I made a translation and another mapping using the third power. This time it is the rear light of a car:


The strong distortions result in rather amorphous pictures. They have not the rather rigid geometric appeal of the kaleidoscopic images. Instead they are more expressionistic.

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